Frequency analysis is necessary and desirable in many applications from radar to spread-spectrum communications. Most often the Fourier transform is used for spectral analysis and frequency domain processing. The continuous time Fourier transform operates on a continuous time signal while discrete Fourier transforms (DFT) operate on samples of a signal: the DFT is a sequence of samples or points equally spaced in frequency. A complex summation of many complex multiplications is required for every sample and imposes a time burden that limits the usefulness of DFT in many applications. For this reason fast Fourier transform systems including fast Fourier transforms (FFT's) and inverse fast Fourier transforms (IFFT's) were developed using mathematical shortcuts to reduce the number of calculations required for DFT's. FFT processing is most often performed in a digital signal processor (DSP) with sixteen bit words because it requires less power and fewer megainstructions per second (MIPS). But accuracy suffers. For example, a sixteen bit (fixed point) 512 sample or point FFT that uses the unconditional shift by one approach has but an eight bit accuracy or approximately a 48 dB SNR. While a thirty-two bit FFT (double precision) has better accuracy, typically more then sixteen bit but requires four to six times the power and operations per multiplication. The rounding off costs about one bit of accuracy per stage of the FFT. The more stages the greater the loss of accuracy. FFTs are typically constructed of butterflies where the number of butterflies in a radix 2 FFT is equal to n/2log2 n where n is the number of samples or points, and the number of stages is log2 n. The pair of equations that make up the two point DFT is called a radix-2 “butterfly”. The butterfly is the core calculation of the FFT. The entire FFT is performed by combining butterflies in patterns determined by the FFT algorithm. So while sixteen bit FFT's are preferred for power and speed they are not satisfactory where accuracy is an important consideration. Another shortcoming of FFT's is that in order to accommodate for overflow conditions an unconditional shift right by one (divide by 2) technique is used. That is, given extreme conditions where both inputs to the butterflies are maximum at 1 the output could be 2, i.e. a 1 with a 1 carry. Since this is intolerable as the outputs must be between 0 and 1, the unconditional shift guarantees that the output doesn't exceed one. However, it pays the price of throwing away another bit in further rounding off. And even further it can be shown that under boundary conditions the maximum output could be 1 +√2 or 2.414 a full 0.414 greater than 2 so that even the unconditional shift with its inherent error does not compensate for the possible overflow error in all conditions. Overall the signal to noise ration SNR is not as high as would be desirable.